Algebra by Thomas W. Hungerford.pdf

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Graduate Texts in Mathematics 73
Editorial Board
S. Axler F.W. Gehring K.A. Ribet

Thomas W. Hungerford
ALGEBRA
Springer

Thomas W. Hungerford
Department of Mathematics
Cleveland State University
Cleveland, OH 44115
USA
Editorial Board
S. Axler
Mathematics Department
San Francisco State
University
San Francisco, CA 94132
USA
axler@sfsu.edu
F.W. Gehring
East Hall
University of Michigan
Ann Arbor, MI 48109
USA
fgehring@
math.Isa.umich.edu
Mathematics Subject Classification (2000): 26-01
Library of Congress Cataloging-in-Publication Data
Hungerford, Thomas W.
Algebra
Bibliography: p.
1. Algebra I. Titie
QA155.H83 512 73-15693
ISBN 0-387-90518-9
ISBN 3-540-90518-9
Printed on acid-free paper.
© 1974 Springer-Verlag New York, Inc.
K.A. Ribet
University of California,
Berkeley
Berkeley, CA 94720-3840
USA
ribet@math.berkeley.edu
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Preface to the
Springer Edition
The reception given to the first edition of Algebra indicates that is has filled a
definite need: to provide a self-contained, one-volume, graduate level algebra text
that is readable by the average graduate student and flexible enough to accomodate
a wide variety of instructors and course contents. Since it has been so well re
ceived, an extensive revision at this time does not seem warranted. Therefore,
no substantial changes have been made in the text for this revised printing. How
ever, aU known misprints and errors have been corrected and several proofs have
been rewritten.
I am grateful to Paul Halmos and F. W. Gehring, and the Springer staff, for
their encouragement and assistance in bringing out this edition. It is gratifying to
know that Algebra will continue to be available to the mathematical community.
Springer-Verlag is to be commended for its willingness to continue to produce
high quality mathematics texts at a time when many other publishers are looking
to less elegant but more lucrative ventures.
Seattle, Washington
June, 1980
THOMAS W. HUNGERFORD
Note on the twelfth printing (2003): A number of corrections were incorporated in the fifth
printingJ thanks to the sharp-eyed diligence of George Bergman and his students at Berkeley and
Keqin Feng of the Chinese University of Science and Technology. Additional corrections appear
in this printing, thanks to Victor Boyko, Bob Cacioppo, Joe L. Mott, Robert Joly, and Joe
Brody.
vii

Preface
Note: A complete discussion of possible
ways of using this text, including sug
gested course outlines, is given on page xv
.
This book is intended to serve as a basic text for an algebra course at the beginning
graduate level. Its writing was begun several years ago when I was unable to find
a one-volume text which I considered suitable for such a course. My criteria for
..suitability," which I hope are met in the present book, are as follows.
(i) A conscious effort has been made to produce a text which an average (but
reasonably prepared) graduate student might read by himself without undue diffi
culty. The stress is on clarity rather than brevity.
(ii) For the reader's convenience the book is essentially self-contained. Con
sequently it includes much undergraduate level material which may be easily omitted
by the better prepared reader.
(iii) Since there is no universal agreement on the content of a first year graduate
algebra course we have included more material than could reasonably be covered in
a single year. The major areas covered are treated in sufficient breadth and depth
for the first year graduate level. Unfortunately reasons of space and economics have
forced the omission of certain topics, such as valuation theory. For the most part
these omitted subjects are those which seem to be least likely to be covered in a one
year course.
(iv) The text is arranged to provide the instructor with maximum flexibility in
the choice, order and degree of coverage of topics, without sacrificing readability
for the student.
(v) There is an unusually large number of exercises.
There are, in theory, no formal prerequisites other than some elementary facts
about sets, functions, the integers, and the real numbers, and a certain amount of
"mathematical maturity." In actual practice, however, an undergraduate course in
modern algebra is probably a necessity for most students. Indeed the book is
written on this assumption, so that a number of concepts with which the typical
graduate student may be assumed to be acquainted (for example, matrices) are
presented in examples, exercises, and occasional proofs before they are formally
treated in the text.
IX

X PREFACE
The guiding philosophical principle throughout the book is that the material
should be presented in the maximum useable generality consistent with good pedago
gy. The principle is relatively easy to apply to various technical questions. It is more
difficult to apply to broader questions of conceptual organization. On the one hand,
for example, the student must be made aware of relatively recent insights into the
nature of algebra: the heart of the matter is the study of morphisms (maps); many
deep and important concepts are best viewed as universal mapping properties. On
the other hand, a high level of abstraction and generality is best appreciated and
fully understood only by those who have a firm grounding in the special situations
which motivated these abstractions. Consequently, concepts which can be character
ized by a universal mapping property are not de
fined via this property if there is
available a definition which is more familiar to or comprehensible by the student.
In such cases the universal mapping property is then given in a theorem.
Categories are introduced early and some tenninology of category theory is used
frequently thereafter. However, the language of categories is employed chiefly as a
useful convenience. A reader who is unfamiliar with categories should have little
difficulty reading most of the book, even as a casual reference. Nevertheless, an
instructor who so desires may give a substantial categorical flavor to the entire course
without ditnculty by treating Chapter X (Categories) at an early stage. Since it is
essentially independent of the rest of the book it may be read at any time.
Other features of the mathematical exposition are as follows.
Infinite sets, infinite cardinal numbers, and transfinite arguments are used routine
ly. All of the necessary set theoretic prerequisites, including complete proofs of
the relevant facts of cardinal arithmetic, are given in the Introduction.
The proof of the Sylow Theorems suggested by R. J. Nunke seems to clarify an
area which is frequently confusing to many students.
Our treatment of Galois theory is based on that of Irving Kaplansky, who has
successfully extended certain ideas of Emil Artin. The Galois group and the basic
connection between subgroups and subfields are defined in the context of an ab
solutely general pair of fields. Among other things this permits easy generalization of
various results to the infinite dimensional case. The Fundamental Theorem is proved
at the beginning, before splitting fields, normality, separability, etc. have been
introduced. Consequently the very real danger in many presentations, namely that
student will lose sight of the forest for the trees, is minimized and perhaps avoided
entirely.
In dealing with separable field extensions we distinguish the algebraic and the
transcendental cases. This seems to be far better from a pedagogical standpoint than
the Bourbaki method of presenting both cases simultaneously.
If one assumes that all rings have identities, all homomorphisms preserve identi
ties and all modules are unitary, then a very quick treatment of semisimple rings
and modules is possible. Unfortunately such an approach does not adequately pre
pare a student to read much of the literature in the theory of noncommutative rings.
Consequently the structure theory of rings (in particular, semisimple left Artinian
rings) is presented in a more general context. This treatment includes the situation
mentioned above, but also deals fully with rings without identity, the Jacobson
radical and related topics. In addition the prime radical and Goldie's Theorem on
semiprime rings are discussed.
There are a large number of exercises of varying scope and difficulty. My experi
ence in attempting to �'star" the more difficult ones has thoroughly convinced me of

PREFACE XI
the truth of the old adage: one man's meat is another's poison. Consequently no
exercises are starred. The exercises are important in that a student is unlikely to
appreciate or to master the material fully if he does not do a reasonable number of
exercises. But the exercises are not an integral part of the text in the sense that non
trivial proofs of certain needed results are left entirely to the reader as exercises.
Nevertheless, most students are quite capable of proving nontrivial propositions
provided that they are given appropriate guidance. Consequently, some theorems
in the text are followed by a "sketch of proof" rather than a complete proof. Some
times such a sketch is no more than a reference to appropriate theorems. On other
occasions it may present the more difficult parts of a proof or a necessary "trick"
in full detail and omit the rest. Frequently aU the major steps of a proof will be
stated, with the reasons or the routine calculational details left to the reader. Some
of these latter "sketches" would be considered complete proofs by many people. In
such cases the word usketch" serves to warn the student that the proof in question
is somewhat more concise than and possibly not as easy to follow as some of the
"complete" proofs given elsewhere in the text.
Seattle, Washington
September, 1973
THOMAS W. HUNGERFORD

Acknowledgments
A large number of people have influenced the writing of this book either directly
or indirectly. My first thanks go to Charles Conway, Vincent McBrien, Raymond
Swords, S.J., and Paul Halmos. Without their advice, encouragement, and assistance
at various stages ofmy educational career I would not have become a mathematician.
I also wish to thank my thesis advisor Saunders Mac Lane, who was my first guide
in the art of mathematical exposition. I can only hope that this book approaches
the high standard of excellence exemplified by his own books.
My colleagues at the University of Washington have offered advice on various
parts of the manuscript. In particular I am grateful to R. J. Nunke, G. S. Monk,
R. Warfield, and D. Knudson. Thanks are also due to the students who have used
preliminary versions of the manuscript during the past four years. Their comments
have substantially improved the final product.
It is a pleasure to acknowledge the help of the secretarial staff at the University
of Washington. Two preliminary versions were typed by Donna Thompson. some
times assisted by Jan Nigh, Pat Watanabe, Pam Brink, and Sandra Evans. The
final version was typed by Sonja Ogle, Kay Kolodziej Martin, and Vicki Caryl,
with occasional assistance from Lois Bond, Geri Button, and Jan Schille.
Mary, my wife, deserves an accolade for her patience during the (seemingly inter
minable) time the book was being written. The final word belongs to our daughter
Anne, age three, and our son Tom, age two, whose somewhat unexpected arrival
after eleven years of marriage substantially prolonged the writing of this book:
a small price to pay for such a progeny.
XIII

Suggestions
on the Use of this Book
GENERAL I N FORMATION
Within a given section all definitions, lemmas, theorems, propositions and corol
laries are numbered consecutively (for example, in section 3 of some chapter the
fourth numbered item is Item 3.4). The exercises in each section are numbered in a
separate system. Cross references are given in accordance with the following
scheme.
(i) Section 3 of Chapter V is referred to as section 3 throughout Chapter V and
as section V.3 elsewhere.
(ii) Exercise 2 of section 3 of Chapter V is referred to as Exercise 2 throughout
section V.3, as Exercise 3.2 throughout the other sections of Chapter V, and as
Exercise V.3.2 elsewhere.
(iii) The fourth numbered item (Definition, Theorem, Corollary, Proposition,
or. Lemma) of section 3 of Chapter Vis referred to as Item 3.4 throughout Chapter V
and as Item V.3.4 elsewhere.
The symbol • is used to denote the end of a proof. A complete list of mathematical
symbols precedes the index.
For those whose Latin is a bit rusty, the phrase mutatis mutandis may be roughly
translated: •'by changing the things which (obviously) must be changed (in order
that the argument will carry over and make sense in the present situation).''
The title "'proposition" is applied in this book only to those results which are nor
used in the sequel (except possibly in occasional exercises or in the proof of other
upropositions..). Consequently a reader who wishes to follow only the main line of
the development may omit all propositions (and their lemmas and corollaries) with
out hindering his progress. Results labeled as lemmas or theorems are almost always
used at some point in the sequel. When a theorem is only needed in one or two
places after its initial appearance, this fact is usually noted. The few minor excep
tions to this labeling scheme should cause little difficulty.
INTERDEPENDENCE OF CHAPTERS
The table on the next page shows chapter interdependence and should be read in
conjunction with the Table of Contents and the notes below (indicated by super
scripts). In addition the reader should consult the introduction to each chapter for
information on the interdt:pendence of the various sections of the chapter.
XV

xvi
l!
II
STRUCTURE
OF GROUPS
lt 2,5
v
GALOIS
THEORY
I 6
VI
STRUCTURE
OF FIELDS
SUGGESTI ONS ON THE USE OF THIS BOOK
INTRODUCTION
v
vn
LINEAR
ALGEBRA
SETS
I
I
GROUPS
l!
III
RINGS
I
I
I
I
V
IV
2,3
2,3
2,3,4
MODULES
v
VIII
COMMUTATIVE
RINGS
&MODULES
7
. -�
1 1
X
CATEGORIES
v 8,9, I 0
IX
STRUCfURE
OF RINGS

SUGG ESTED COURSE OUTLI N ES xvii
NOTES
1. Sections 1-7 of the Introduction are essential and are used frequently in the
sequel. Except for Section 7 (Zorn's Lemma) this material is almost all elementary.
The student should also know a definition of cardinal number (Section 8, through
Definition 8.4). The rest of Section 8 is needed only five times. (Theorems Il.1.2 and
IV.2.6; Lemma V.3.5; Theorems V.3.6 and Vl.1.9). Unless one wants to spend a
considerable amount of time on cardinal arithmetic, this material may well be
postponed until needed or assigned as outside reading for those interested.
2. A student who has had an undergraduate modern algebra course (or its
equivalent) and is familiar with the contents of the Introduction can probably begin
reading immediately any one of Chapters I, Ill, IV, or V.
3. A reader who wishes to skip Chapter I is strongly advised to scan Section
I.7 to insure that he is familiar with the language of category theory introduced
there.
4. With one exception, the only things from Chapter III needed in Chapter IV
are the basic definitions of Section 111.1. However Section 111.3 is a prerequisite for
Section IV.6.
5. Some knowledge of solvable groups (Sections II.7, 11.8) is needed for the
study of radical field extensions (Section V.9).
6. Chapter VI requires only the first six sections of Chapter V.
7. The proof of the Hilbert Nullstellensatz (Section VIII.7) requires some
knowledge of transcendence degrees (Section VI.l) as well as material from Section
V.3.
8. Section VIlLI (Chain Conditions) is used extensively in Chapter IX, but
Chapter IX is independent of the rest of Chapter VIII.
9. The basic connection between matrices and endomorphisms of free modules
(Section VII.I, through Theorem VII.I.4) is used in studying the structure of rings
(Chapter IX).
I0. Section V.3 is a prerequisite for Section IX.6.
11. Sections 1.7, IV.4, and IV.5 are prerequisites for Chapter X; otherwise
Chapter X is essentially independent of the rest of the book.
SUGGESTED COU RSE OUTLINES
The information given above, together with the introductions to the various chapters,
is sufficient for designing a wide variety of courses of varying content and length.
Here are some of the possible one quarter courses (30 class meetings) on specific
topics.
These descriptions are somewhat elastic depending on how much is assumed, the
level of the class, etc. Under the heading Review we list background material (often
of an elementary nature) which is frequently used in the course. This material may

XVIII SUGGESTIONS ON THE USE OF THI S BOOK
be assumed or covered briefly or assigned as outside reading or treated in detail if
necessary� depending on the background of the class. It is assumed without ex
plicit mention that the student is familiar with the appropriate parts of the Intro
duction (see note 1, p. xvii). Almost all of these courses can be shortened by omit
ting all Propostions and their associated Lemmas and Corollaries (see page xv).
GROUP THEORY
Review: Introduction, omitting most of Section 8 (see note 1, p. xvii). Basic
Course: Chapters I and II, with the possible omission of Sections 1.9, 11.3 and the
last halfof II.7. It is also possible to omit Sections 11.1 and 11.2 or at least postpone
them until after the Sylow Theorems (Section 11.5).
MODULES AND THE STRUCTURE OF RINGS
Review: Sections 111.1 and 111.2 (through Theorem 111.2.13). Basic Course: the
rest of Section 111.2; Sections 1-5 of Chapter IV1; Section VII.I (through Theorem
VII. 1.4); Section VIII. I; Sections 1-4 of Chapter IX. Additional Topics: Sections
111.4, IV.6, IV.7, IX.5; Section IV.5 if not covered earlier; Section IX.6; material
from Chapter VIII.
FIELDS AND GALOIS THEORY
Review: polynomials, modules, vector spaces (Sections 111.5, 111.6, IV.1, IV.2).
Solvable groups (Sections II.7, 11.8) are used in Section V.9. Basic Course2: Sec
tions 1-3 of Chapter V, omitting the appendices; Definition V.4.1 and Theorems
V.4.2 and V.4.12; Section V.5 (through Theorem 5.3); Theorem V.6.2; Section
V.7, omitting Proposition V.7.7-Corollary V.7.9; Theorem V.8.1; Section V.9
(through Corollary V.9.5); Section VI. I. Additional Topics: the rest of Sections
V.5 and V.6 (at least through Definition V.6.10); the appendices to Sections V.1-
V.3; the rest of Sections V.4, V.9, and V.7; Section V.8; Section VI.2.
LINEAR ALGEBRA
Review: Sections 3-6 of Chapter III and Section IV.1; selected parts of Section
IV.2 (finite dimensional vector spaces). Basic Course: structure of torsion mod
ules over a PID (Section IV.6, omitting material on free modules); Sections 1-5 of
Chapter VII, omitting appendices and possibly the Propositions.
11f the stress is primarily on rings, one may omit most of Chapter IV. Specifically, one
need only cover Section IV.1; Section IV.2 (through Theorem IV.2.4); Definition IV.2.8;
and Section IV.3 (through Definition IV.3.6).
2The outline given here is designed so that the solvability of polynomial equations can be
discussed quickly after the Fundamental Theorem and splitting fields are presented; it re
quires using Theorem V.7.2 as a definition, in place of Definition V.7.1. The discussion may
be further shortened if one considers only finite dimensional extensions and omits algebraic
closures, as indicated in the note preceding Theorem V.3.3.

SUGGESTED COU RSE OUTLIN ES XIX
COMMUTATIVE AlGEBRA
Review : Sections 111.1,111.2 (through Theorem 111.2.13). Basic Course: the rest of
Section 111.2; Sections 111.3 and 111.4; Section IV.l; Section IV.2 (through Corollary
IV.2.2); Section IV.3 (through Proposition IV.3.5); Sections 1-6 of Chapter VIII,
with the possible omission of Propositions. Additional topics : Section VIII.7
(which also requires background from Sections V.3 and VI. I).

Table of Contents
Preface . . . . . . . . . _ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . . . 0 • • • • • • • • • 0 0 0 • • • • • • • • 0 • • • • • 0
Suggestions on the Use of This Book . 0 • • • • • • • • • • • 0 0 • 0 0 • • 0 • • • 0 • • •
.
IX
Xlll
XV
Introduction: Prerequisites and Preliminaries. . . . . . . . . . . I
1. Logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 1
2. Sets and Classes . . . . 0 • • • • • • • 0 • 0 • • • • • • • • • • • • • • • • 0 • • • • • • • • • • • 1
3. Functions . . . . 0 • 0 0 • • 0 • • • • 0 0 0 • • • • • • • • 0 0 0 • • • 0 • • • 0 0 0 • • • • 0 0 0 • • 0 3
40 Relations and Partitions . . . . 0 • • • • • 0 • • • • • • • • • • • • • • • 0 • • • • 0 • • • • 6
5. Products . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
6. The Integers.. . . 0 • 0 • 0 • • • • • • • • • 0 • • • • • 0 • • • • _ • • • 0 • • • • • • • • • 0 • • • 9
7. The Axiom of Choice,Order and Zorn's Lemma . . . 0 • • • • 0 • 0 • • 0 • 12
80 Cardinal Numbers. 0 • • • • • 0 0 0 0 0 • • • 0 • 0 • • • • 0 • • 0 0 • 0 • • 0 0 • • 0 • • 0 • • 0 15
Chapter 1: Groups. . _ . 0 • _ 0 • • • • • • • • • 0 • • 0 • • • • • • • • 0 • • • • 0 • • • • • • 23
1. Semigroups, Monoids and Groups. . 0 • • • • • 0 0 • 0 0 • • • • 0 • • • • • • 0 • • 24
2. Homomorphisms and Subgroups. 0 0 0 • • • • • • • • • • • • • • • • • • • 0 • • • • • 30
3. Cyclic Groups 0 0 0 • • • • • • • • • • 0 0 • 0 0 • • • • • • • 0 • • • • • 0 • • • • 0 • • • 0 • • 0 • 35
4. Cosets and Counting . . . 0 • 0 • • 0 • • • • 0 • • • • • 0 0 • 0 • 0 • • • • 0 • • 0 • 0 0 0 • • 37
5° Normality,Quotient Groups,and Homomorphisms. . . . . . . . . 0 0 . 0 41
6. Symmetric, Alternating,and Dihedral Groups. . 0 • • • • 0 • • • • • 0 0 • • 0 46
10 Categories: Products,Coproducts,and Free Objects . . . . . . 0 • • • • 0 52
8. Direct Products and Direct Sums. 0 • • 0 0 • • • 0 • • • • • • 0 • • • • • • 0 • • • • • 59
9. Free Groups, Free Products, Generators & Relations . . . . . . . . . 0 0 64
Chapter II: The Structure of Groups_ . . . _ _ . . . . 0 • • • _ • • • 0 • • • 70
1. Free Abelian Groups . . . . . . . . . . . 0 0 • 0 • • • 0 • • • • • • • • • • • • • • • • • • 0 • 70
20 Finitely Generated Abelian Groups. . . . . . . . . . . . 0 0 • • • 0 • • 0 _ • • 0 • •
76
xxi

XXII TABLE OF CONTENTS
3. The Krull-Schmidt Theorem.. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4. The Action of a Group on a Set.. . . . . .. . . . . . . . . . . . . . ... . . . . . 88
5. The Sylow Theorems. . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . 92
6. Classification of Finite Groups.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7. Nilpotent and Solvable Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8. Norm��tl and Subnormal Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Chapter Ill: Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . . 114
1. Rings and Homomorphisms. . . . . . . . 0 • • • • • • • • • • • • • • • • • • • • • • • • 115
2. Ideals.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .. . . . . . . . . . . . 122
3. Factorization in Commutative Rings. . . . . . . . . . 0 • • • • • • • • • • • • • • • 135
4. Rings of Quotients and Localization.0 • • • • • • • • • • • • • • • • • • • • • • • • 142
5. Rings of Polynomials and Formal Power Series...... . . . . . . . . . . 149
6. Factorization in Polynomial Rings. 0 • • • • • • • • • • • • • • • • • • • • • • • • • 157
Chapter IV: Modules . . . . . 0 . 0 • • • • • • • • • • • 0 . . . . . . . . . . . . . . . . . . 168
1. Modules, Homomorphis ms and Exact Sequences . . . . . . . . . . . 169
2. Free Modules and Vector Spaces. . . . . . . . . . . . . . . . . . . . . . 180
3. Projective and Injective Modules . . . . . . . . 0 • • • 0 • • • 0 • • 0 • • 190
4. Hom and Duality . . . . . . . 0 • • • • • • • • • • • • • • • • • • 0 • • • • • 199
5. Tensor Products. 0 • • • • • • • • • • 0 0 0 0 • 0 0 0 • • 0 • • • 0 • 0 0 • • • 207
6. Modules over a Principal Ideal Domain 0 • 0 • • • • • • • 0 • • • • • • • 218
7. Algebras 0 • • 0 • • • • o • • • • • o • • • • • • • • • • • • • • • • • • • • • • • 226
Chapter V: Fields and Galois Theory . . . . . . .. . .. . . . . 0 • • • 0 • 230
1. Field Extensions . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Appendix: Ruler and Compass Constructions. . . . . . . . . . . . . .. 238
2. The Fundamental Theorem. o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Appendix: Symmetric Rational Functions... . .. . . . . .. . . . . . .. 252
3. Splitting Fields, Algebraic Closure and Normality. . . . . . . . . . . . . . 257
Appendix: The Fundamental Theorem of Algebra............ 265
4. The Galois Group of a Polynomial.. . . . . . . . . . . . . . . . . . . . . . . . . . 269
5. Finite Fields. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
6. Separability_· .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 282
7. Cyclic Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 289
8. Cyclotomic Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
9. Radical Extensions. . . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . 302
Appendix: The General Equation of Degree n. . . . . . . . . . . . . . . 307
Chapter VI: The Structure of Fields. . . . . . . . . . . . . . . . . . . . . . . 311
1. Transcendence Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
2. Linear Disjointness and Separability . . . . . _ . . . . . _ . . _ . _ . . _ . . . . . . 318

TABLE OF CONTENTS xx111
Chapter VII: Linear Algebra.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
1. Matrices and Maps. . . . . . . . . . . . . .. . . . .. . . . . . . .. . . . . . . .. . . . . . 328
2. Rank and Equivalence. . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . 335
Appendix: Abelian Groups Defined by
Generators and Relations . . . . . . . . . . .. . . . . . . . . .. . 343
3. Determinants... . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
4. Decomposition of a Single Linear Transformation and Similarity . 355
5. The Characteristic Polynomial, Eigenvectors and ·Eigenvalues. . . . 366
Chapter VIII: Commutative Rings and Modules.... . . . . . . . 371
1. Chain Conditions. . . . . . . . . . . . . . . . . .. . . .. .. . . . . . . . . . . . . . . . . . 372
2. Prime and Primary Ideals. . . . . .... . . . . . . . . . . . . . . . . . . . . . . . . .. 377
3. Primary Decomposition.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
4. Noetherian Rings and Modules.. . . . . . . . . . . ... . ... . . . . . . . .. .. 387
5 . Ring Extensions.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . 394
6. Dedekind Domains. . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . 400
7. The Hilbert Nullstellensatz.. . . . . . ... . . . . . . . . . . . .. . . . . . . . . . . . 409
Chapter IX: The Structure of Rings... .. . . . . . . . . . . . . . . . . . . 414
1. Simple and Primitive Rings.. . . . . . . .. . . .. . . . . .. . . . . . . . . . . . . . . 415
2. The Jacobson Radical. .... . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . 424
3. Semisimple Rings.. . . . . . . . . .. . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . 434
4. The Prime Radical; Prime and Semiprime Rings . . . . . . .. . . . . . . . 444
5. Algebras. . .. . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . ... . . . ... . . .. . . . 450
6. Division Algebras .... .. . .. . . . _ . . . . . . . . . . • .. . .. .. . . . . . . . . . . . 456
Chapter X: Categories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
1. Functors and Natural Transformations. . . . .. . . .. . . . . ... . . . . . . 465
2. Adjoint Functors. .. . . . . . .... . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 476
3. Morphisms. .. . . . . .. . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . 480
List of Symbols .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .. . . .. . . .. . 485
Bibliography.. . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . .. . . . . . . .... . . . . ... . 489
Index . . . - - . . . . . .. . . . . . . . . . . ... . . . ... . ..... .. . - . . . . . . . . . . . . . . . . . . 493

INTRODUCTION
PREREQUISITES AN D
PRELI MI NARIES
In Sections 1-6 we summarize for the reader's convenience some basic material with
which he is assumed to be thoroughly familiar (with the possible exception ofthe dis
tinction between sets and proper classes (Section 2), the characterization of the
Cartesian product by a universal mapping property (Theorem5.2) and the Recursion
Theorem 6.2). The definition of cardinal number (first part ofSection 8) will be used
frequently. The Axiom of Choice and its equivalents (Section 7) and cardinal arith
metic (last part of Section 8) may be postponed until this information is actually
used. Finally the reader is presumed to have some familiarity with the fields Q, R,
and C of rational, real, and complex numbers respectively.
1. LOGIC
We adopt the usual logical conventions, and consider only statements that have a
truth value of either true or false (not both). If P and Q are statements, then the
statement "'P and Q'' is true ifboth P and Q are true and false otherwise. The state
ment "P or Q" is true in allcases except when both P and Q are false. An implication
is a statement of the form "P implies Q" or "if P, then Q" (written symbolically as
P ::=} Q). An implication is false ifP is true and Q is false; it is true in all other cases.
In particular, an implication with a false premise is always a true implication. An
equivalence or biconditional is a statement of the form "P implies Q and Q im
plies P." This is generally abbreviated to "'P ifand only if Q" (symbolically P <::::} Q).
The biconditional "'P � Q" is true exactly when P and Q are both true or both
false; otherwise it is false. The negation ofthe statement P is the statement "'it is not
the case that P." It is true if and only if P is false.
2. SETS AND CLASSES
Our approach to the theory of sets will be quite informal. Nevertheless in order
to define adequately both cardinal numbers (Section 8) and categories (Section I.7) it
1

2 PREREQUISITES AND PRELI M I NAR I ES
will be necessary to introduce at least the rudiments of a formal axiomatization of
set theory. In fact the entire discussion may, if desired, be made rigorously precise;
see Eisenberg [8] or Suppes [10). An axiomatic approach to set theory is also useful in
order to avoid certain paradoxes that are apt to cause difficulty in a purely intuitive
treatment of the subject. A paradox occurs in an axiom system when both a state
ment and its negation are deducible from the axioms. This in turn implies (by an
exercise in elementary logic) that every statement in the system is true, which is
hardly a very desirable state of affairs.
In the Gooel-Bernays form of axiomatic set theory, which we shall follow, the
primitive (undefined) notions are class, membership, and equality. Intuitively we con
sider a class to be a collectionA of objects (elements) such that given any object x it
is possible to determine whether or not x is a member (or element) ofA. We write
x E A for "x is an element ofA" and x +A for ''x is not an element ofA:' The axioms
are formulated in terms of these primitive notions and the first-order predicate
calculus (that is, the language of sentences built up by using the connectives and,
or, not, implies and the quantifiers there exists and for all). For instance, equal
ity is assumed to have the following properties for all classes A, B, C: A = A;
A = B => B = A; A = B and B = C => A = C; A = B and x E A => x E B. The
axiom of extensionality asserts that two classes with the same elements are equal
(formally, [x E A {=> x E B) =:::) A = B).
A classA is defined to be a set if and only if there exists a class B such thatA e B.
Thus a set is a particular kind ofclass. A class that is not a set is called a proper class.
Intuitively the distinction between sets and proper classes is not too clear. Roughly
speaking a set is a "small" class and a proper class is exceptionally "large." The
axiom of class formation asserts that for any statement P(y) in the first-order predi
cate calculus involving a variable y, there exists a classA such that x E A if and only
if x is a set and the statement P(x) is true. We denote this class A by l x I P(x)J, and
refer to uthe class of all X such that P(x)." Sometimes a class is described simply by
listing its elements in brackets, for example, {a,b,c}.
EXAMPLE.1 Consider the classM = {X I X is a set and X .X}. The statement
X fX is not unreasonable since many sets satisfy it (for example, the set ofall books is
not a book).Mis a proper class. For if Mwere a set, then eitherMeMorM4M.
But by the definition ofM, MeMimpliesMtMand M¢MimpliesMeM. Thus in
either case the assumption thatM is a set leads to an untenable paradox:MeM
MdMt M
.
We shall now review a number offamiliar topics (unions, intersections, functions,
relations, Cartesian products, etc.). The presentation will be informal with the men
tion of axioms omitted for the most part. However, it is also to be understood that
there are sufficient axioms to guarantee that when one of these constructions is per
formed on sets, the result is also a set (for example, the union of sets is a set; a sub
class of a set is a set). The usual way of proving that a given class is a set is to show
that it may be obtained from a set by a sequence of these admissible constructions.
A class A is a subclass of a class B (written A c B) provided:
for all x eA, x eA =:::) x E B. (1)
1This was first propounded (in somewhat different form) by Bertrand Russell in 1902 as
a paradox that indicated the necessity of a formal axiomatization of set theory.

3. FUNCTION S
By the axioms of extensionality and the properties of equality:
A = B ¢::} A c B and B c A.
3
A subclass A ofa class B that is itselfa set is called a subset ofB. There are axioms to
insure that a subclass of a set is a subset.
The empty set or null set (denoted 0) is the set with no elements (that is, given
any x, x t 0). Since the statement "x E 0" is always false, the implication (1) is al
ways true when A = 0. Therefore 0 C B for every classB. A is said to be a proper
subclass of B if A c B but A � 0 and A � B.
The power axiom asserts that for every set A the class P(A) ofall subsets of A is
itself a set. P(A) is called the power set ofA; it is also denoted 2A.
A family of sets indexed by (the nonempty class) I is a collection of sets A,, one
for each i E I (denoted {Ai I i E I}). Given such a family, its union and intersection are
defined to be respectively the classes
UA; = {x I x E Ai for some i E JJ; and
iel
nA. = {X I X E A, for every i E I}.
iel
If I is a set, then suitable axioms insure that UAi and nA. are actually sets. If
iei iel
I = { 1,2, . . . , nJ one frequently writes At U A2 U - · · U An in place of UA, and
ial
similarly for intersections. If A n B = 0, A and B are said to be disjoint.
If A and Bare classes, the relative complement of A in B is the following subclass
of B:
B - A = {x I x E B and x � AJ.
Ifall the classes under discussion are subsets of some fixed set U (called the universe
of discussion), then U- A is denoted A' and called simply the complement of A.
The reader should verify the following statements.
A n (UBi) = U<A n Bi) and
ial iel
A U (nBi) = n(A U B;).
iei iei
(2)
(UAi)' = nA/ and <nAi)' = UA/ (DeMorgan's Laws). (3)
iEl iei iei iei
A U B = B � A C B ¢::> A n B = A. (4)
3. FUNCTIONS
Given classes A and B, a function (or map or mapping) [from A to B (written
f: A --+B) assigns to each a E A exactly one element b E B; b is called the value ofthe
function at a or the image of a and is usually written f(a). A is the domain of the
function (sometimes written Domfl and B is the range or codomain. Sometimes it is
convenient to denote the effect ofthe functionf on an element ofA by a f-. f(a). Two
functions are equal if they have the same domain and range and have the same
value for each element of their common domain.

4 PREREQU ISITES AN D PRELI M I NARI ES
Iff :A--+ B is a function and S c A, the function from S to B given by
a� f(a), for a e:S
is called the restriction of f to S and is denoted f I S : S ---+ B. IfA is any class, the
identity function onA (denoted lA : A ---+A) is the function given by a� a. IfS C A,
the function lA I S : S--+A is called the inclusion map of S into A.
Let f: A ---+B and g : B--. C be functions. The composite offand g is the function
A--. C given by
a� g(f(a)), a eA.
The composite function is denoted g o f or simply g f. If h : C--. D is a third function,
it is easy to verify that h(gf) = (hg)f. Iff : A ---+ B, then/o 1A = f = lB o f : A---+ B.
A diagram of functions:
is said to be commutative if gf = h. Similarly, the diagram:
is commutative ifkh = gf. Frequently we shall deal with more complicated diagrams
composed of a number of triangles and squares as above. Such a diagram is said to
be commutative if every triangle and square in it is commutative.
Let f: A--. B be a function. If S c A, the image ofS under f (denoted f(S)) is
the class
{ b eB I b = f(a) for some a eSJ.
The class f(A) is called the image off and is sometimes denoted Im f If T c B, the
inverse image of T under/(denoted f-1(T)) is the class
{ a eA I f(a) e T}.
If T consists of a single element, T = { bl, we write /-1(b) in place of/ -1( D. The
following facts can be easily verified:
for S C A,/-1(/(S)) ::> S;
for T c B, f( /-1(D) c T.
For any family { Ti I i eIl of subsets of B,
/-1(UTi) = U /-1(Ti);
iP:.l iP:.l
f-1(nTi) = nf-I( Ti).
i,p;J iP:.l
A function f: A--+ B is said to be injective (or one..to-one) provided
for all a,d e A, a y6. a' ::::::} f(a) � f(a');
(5)
(6)
(7)
(8)
'
'

3. FU NCTI ONS 5
alternatively, fis injective if and only if
for all a,a' EA, f(a) = /(a') => a = a'.
A function fis surjective (or onto) provided f(A) = B; in other words,
for each bE B, b = f(a) for some aE A.
A function fis said to be bijective (or a bijection or a one-to-one correspondence) if it
is both injective and surjective. It follows immediately from these definitions that for
any class A, the identity map IA :A --4 A is bijective. The reader should verify that
for maps f:A --4 B and g : B--+ C,
f and g injective => g fis injective; (9)
fand g surjective => g fis surjective; (10)
gfinjective => /is injective; (11)
g fsurjective => g is surjective. (12)
Theorem 3. 1. Let f: A--+ B be af
unction, with A nonempty.
(i) f is injective if and only if there is a map g : B --4 A such that gf = 1A·
(ii) IfAis a set, then fis surjective ifand only ifthere is a map h : B --4 A such that
fh = lB.
PROOF. Since every identity map is bijective, ( l l) and {12) prove the implica
tions {<=) in {i) and (ii). Conversely if/is injective, then for each bE f(A) there is a
unique aE A with f(a) = b. Choose a fixed aoE A and verify that the map g : B --4 A
defined by
(b) =
{a if b E/(A) and f(a) = b
g
ao if b 4/(A)
is such that g f= 1A. For the converse of {ii) suppose fis surjective. Then f-1{b) C A
is a nonempty set for every bE B. For each bE B choose abE f-1(b) {Note: this re
quires the Axiom ofChoice; see Section 7). Verify that the map h : B --4 A defined by
h(b) = a, is such that fh = ln. •
The map g as in Theorem 3.1 is called a left inverse of fand h is called a right in
verse of f. If a map f: A --+ B has both a left inverse g and a right inverse h, then
g = glB = g( fh) = (gf)h = IAh = h
and the map g = h is called a two-sided inverse of f. This argument also shows that
the two-sided inverse of a map {if it has one) is unique. By Theorem 3.1 if A is a set
and f: A --.. B a function, then
/is bijective <=> /has a two-sided inverse.2 (13)
The unique two-sided inverse of a bijection fis denoted j-1; clearly fis a two-sided
·inverse ofJ-• so that J-• is also a bijection.
'(13) is actually true even when A is a proper class; see Eisenberg [8; p. 146].

6 PREREQU ISITES AND PRELIMI NARI ES
4. RELATIONS AND PARTITIONS
The axiom of pair formation states that for any two sets [elements] a,b there is a
set P = { a,bJ such that x E P if and only if x = a or x = b; if a = b then P is the
singleton { a}. The ordered pair (a,b) is defined to be the set { { a}, { a,b}}; its first com
ponent is a and its second component is b. It is easy to verify that (a,b) = (a',b') ifand
only if a = a' and b = b'. The Cartesian product of classes A and B is the class
A X B = {(a,b) I a E A, b E B} .
Note that A X 0 = 0 = 0 X B.
A subclass R ofA X B is called a relation on A X B. For example, iff : A -4 B is
a function, the graph of/is the relation R = { (a,f(a)) I a e A } . Since/is a function,
R has the special property:
every element of A is the first component of
one and only one ordered pair in R.
(14)
Conversely any relation R on A X B that satisfies (14), determines a unique function
f : A -4 B whose graph is R (simply define f(a) = b, where (a,b) is the unique
ordered pair in R with first component a). For this reason it is customary in a formal
axiomatic presentation of set theory to identify a function with its graph, that is, to
define a function to be a relation satisfying (14). This is necessary, for example, in
order to prove from the axioms that the image of a set under a function is in fact
a set.
Another advantage ofthis approach is that it permits us to define functions with
empty domain. For since 0 X B = 0 is the unique subset of 0 X B and vacuously
satisfies (14), there is a unique function 0 -4 B. It is also clear from (14) that there
can be a function with empty range only ifthe domain is also empty. Whenever con
venient we shall think of a function as a relation satisfying (14).
A relation R on A X A is an equivalence relation on A provided R is:
reflexive : (a,a) E R for all a E A;
symmetric: (a,b) E R � (b, a) E R;
transitive : (a,b) E R and (b,c) E R � (q,c) E R.
(15)
(16)
(17)
If R is an equivalence relation on A and (a,b) E R, we say that a is equivalent to b
under R and write a 1"./ b or aRb; in this notation (15)-(17) become:
a ,..._, a;
a ,..._, b � b ,..._, a;
a 1"./ b and b 1"./ c =::} a ,..._, c.
(15')
(16')
(17')
Let R (,..._,) be an equivalence relation on A. If a E A, the equivalence class of a
(denoted ii) is the class of all those elements of A that are equivalent to a; that is,
li = { b E A I b ,..._, a}. The class of all equivalence classes in A is denoted A/R and
called the quotient class of A by R. Since R is reflexive, a E li for every a E A ; hence
a � 0, for every a E A ; and if A is a set
Ua = A = U a.
a�A ae.A/R
(18)
(19)

5. PRODUCTS 7
Also observe that
a = b � · a I"J b; (20)
for if a = b, then a e a ==:} a e b ==:} a � b. Conversely, if a � b and c e a, then c � a
and a � b � c I"J b � c e b. Thus a C b; a symmetric argument shows that li C a
and therefore a = b. Next we prove:
for a,b E A' either a n h = 0 or a = li. (21)
If a n b � 0, then there is an element c E a n b. Hence c � a and c � b. Using
symmetry, transitivity and (20) we have: a � c and c ,-.....; b => a ,-.....; b ==:} a = b.
Let A be a nonempty class and {Ai I i e I} a family of subsets of A such that:
Ai � 0, for each i e I;
U A" = A;
iel
Ai n A; = 0 for all i � j E I;
then { Ai I i e I} is said to be a partition of A.
Theorem 4.1. IfA is a nonempty set, then the assignment R � A/R defines a bijec
tion from the set E(A) ofall equivalence relations on A onto the set Q(A) ofallparti
tions ofA.
SKETCH OF PROOF. If R is an equivalence relation on A, then the set A/R
ofequivalence classes is a partition of A by (18), (19), and (21) so that R � A/R de
fines a function f: E(A) --4 Q(A). Define a function g : Q(A) --4 E(A) as follows. If
S = { Ai I i e I} is a partition ofA, letg(S) be the equivalence relation on A given by:
a � b <=> a e Ai and b e Ai for some (unique) i e I. (22)
Verify that g(S) is in fact an equivalence relation such that a = Ai for a e Ai. Com
plete the proof by verifying that f
g = 1ocA> and gf= 1EcA>· Then f is bijective
by (13). •
5. PRODUCTS
Note. In this section we deal only with sets. No proper classes are involved.
Consider the Cartesian product oftwo sets At X A2• An element of At X A2 is a
pair (a1,a2) with ai e Ai, i = 1,2. Thus the pair (at,a2) determines a function /: { 1,2}
--4 A1 U A2 by: /(1) = a., f(2) = a2. Conversely, every function[: { 1,2} --4 A1 U A2
with the property that /(1) e A1 and /(2) e A2 determines an element (a�,a2) =
(/(1),/(2)) of At X A2. Therefore it is not difficult to see that there is a one-to-one
correspondence between the set of all functions of this kind and the set A1 X A2•
This fact leads us to generalize the notion of Cartesian product as follows.
Definition 5.1. Let { Ai I i e I} be a f
amily ofsets indexed by a (nonempty) set I. The
(Cartesian) product ofthe sets Ai is the set ofall functions f : I --4 U Ai such that
ial
f(i) E Ai for all i e I. It is denoted II Ai.
ial

8 PREREQUISITES AN D PRELI M I NAR I ES
If I = {1,2, . . . , n}, the product IT A; is often denoted by A1 X A2 X · · · X An
i£1
and is identified with the set ofall ordered n-tuples (aha2, . . . , an), where a; e A; for
i = 1,2, . . . , n just as in the case mentioned above, where I = {1 ,2}. A similar
notation is often convenient when I is infinite. We shall sometimes denote the
functionfe IT A; by (a;}1ei or simply {a;}, wheref(i) = a; e A; for each i e I.
i£1
If some A1 = 0, then II Ai = 0 since there can be no function f: I --4 U Ai
i!.l
such that /U) e Ai.
If l Ai I i e I} and tBi I i e I} are families of sets such that Bi c Ai for each i e I,
then every function l --4 U Bi may be considered to be a function l--4 U Ai. There-
�� I
fore we consider II Bi to be a subset of II Ai.
i,] iF.]
Let II Ai be a Cartesian product. For each k e I define a map 7rk : II Ai � Ak
iF.] iF.J
by /� f(k), or in the other notation, { ad � ak. 7rk is called the (canonical) projec-
tion of the product onto its kth component (or factor). If every Ai is nonempty, then each
1T,_ is surjective (see Exercise 7.6).
The product II Ai and its projections are precisely what we need in order
iel
to prove:
Theorem 5.2. Let { Ai I i e I } be afamily ofsets indexed by I. Then there exists a set
D, together with a f
amily ofmaps { 7ri : D ----. Ai I i e I} with thef
ollowingproperty: for
any set Candf
amily ofmaps { 'Pi : C ----. Ai I i e Il , thereexists a uniquemap q; : C ----. D
such that 1ri'P = 'Pi/
or all i E L Furthermore, D is uniquely determinedup to a bijection.
The last sentence means that if D' is a set and {1r/ : D' ------+ A, I i e /} a family of
maps, which have the same property as D and { 1ri} , then there is a bijection D ----. D'.
PROOF OF 5.2. (Existence) Let D = II Ai and let the maps 1ri be the projec
iEI
tions onto the i
th components. Given C and the maps 'Pi-r define q; : C ------+ II Ai by
i,]
c l---+ /c, where fc(i) = <Pi(c) e Ai. It follows immediately that 1ri<P = 'Pi for all i e I. To
show that q; is unique we assume that q;' : C -+ II Ai is another map such that
i£1
1ri'P' = 'Pi for all i e I and prove that 'P = <P
'
. To do this we must show that for each
c e C, (f?(c), and q;'(c) are the same element of II Ai - that is, <P(c) and q;'(c) agree as
it!I
functions on J: ('P(c))(i) = (q;'(c)Xi) for all i e I. But by hypothesis and the definition
of 1ri we have for every i e 1:
(q;'(c))(i) = 1ri'P'(c) = <Pi(c) = /c(i) = (<P(c))(i).
(Uniqueness) Suppose D' (with maps 1r/ : D' -+ Ai) has the same property as
D = II Ai. If we apply this property (for D) to the family of maps { 1r/ : D' ------+ Ai}
ial
and also apply it (for D') to the family { 7ri : D -+ Ai l , we obtain (unique) maps
'
l

6. THE I NTEGERS 9
'P : D' --+ D and 1/1 : D --+ D' such that the following diagrams are commutative for
each i e !:
'
Combining these gives for each i e I a commutative diagram
Thus 'Pl/1 : D --+ D is a map such that 7r"i{'{Jl/l) = 1ri for all i e !. But by the proofabove,
there is a unique map with this property. Since the map ln : D --+ D is also such that
1rilD = 1ri for all i e /, we must have 'Pl/1 = 1n by uniqueness. A similar argument
shows that 4''P = ln'· Therefore, 'P is a bijection by (13) and D = II Ai is uniquely
determined up to a bijection. •
ie.I
Observe that the statement of Theorem 5.2 does not mention elements; it in
volves only sets and maps. It says, in effect, that the product II Ai is characterized
iel
by a certain universal mapping property. We shall discuss this concept with more pre-
cision when we deal with categories and functors below.
6. THE INTEGERS
We do not intend to give an axiomatic development of the integers. Instead we
assume that the reader is thoroughly familiar with the set Z of integers, the set
N = {0,1 ,2, . . . } of nonnegative integers (or natural numbers) the set N* = { 1 ,2, . . . J
of positive integers and the elementary properties of addition, multiplication, and
order. In particular, for all a,b,c e Z:
(a + h) + c = a + (b + c) and (ab)c = a(bc) (associative laws); (23)
a + b = b + a and ah = ba (commutative laws); (24)
a(b + c) = ab + ac and (a + b)c = ac + be (distributive laws); (25)
a + 0 = a and al = a (identity elements); (26)
for each a e Z there exists -a e Z such that a + (-a) = 0 (additive inverse);
(27)
we write a - h for a + (-b).
ab = 0 <=> a = 0 or b = 0;
a < b ==:) a + c < h + c for all c e Z;
a < b =::} ad < bd for all d e N*.
(28)
(29)
(30)
We write a < b and b > a interchangeably and write a < b if a < b or a = b. The
absolute value Ia! of a e Z is defined to be a if a > 0 and -a if a < 0. Finally we
assume as a basic axiom the

10 PREREQU ISITES AND PRELI M I NAR I ES
Law of Well Ordering. Every nonempty subset S of N contains a least element (that
is, an element b e S such that b < c for all c e S).
In particular, 0 is the least element of N.
In addition to the above we require certain facts from elementary number theory,
some of which are briefly reviewed here.
Theorem 6.1. (Principle ofMathematical Induction) IfS is a subset ofthe set N of
natural numbers such that 0 e S and either
or
(i) n e S =::} n + J e S f
or all n e N;
(ii) m e S f
or all 0 < m < n � n e S for all n e N;
then S = N.
PROOF. If N - S � fZ), let n ¢ 0 be its least element. Then for every m < n,
we must have m * N - S and hence m e S. Consequently either (i) or (ii) implies
n e S, which is a contradiction. Therefore N - S = 0 and N = S. •
REMARK. Theorem 6.1 also holds with 0, N replaced by c, Me = {x e Z I x > c l
for any c e Z.
In order to insure that various recursive or inductive definitions and proofs in the
sequel (for example, Theorems 8.8 and 111.3.7 below) are valid, we need a technical
result:
Theorem 6.2. (Recursion Theorem) lfS is a set, a e S and for each n e N, fn : S --4 S is
afunction, then thereis a uniquefunction <P : N --4 S such that <P(O) = a and <P(n + 1) =
fn(<P(n)) f
or every n e N.
SKETCH OF PROOF. We shall construct a relatio� R on N X S that is the
graph of a function <P : N --4 S with the desired properties. Let g be the set of all
subsets Y of N X S such that
(O,a) e Y; and (n,x) e Y � (n + l,.fn(x)) e Y for all n e N.
Then g � f2) since N X S e g. Let R = n Y; then R e g. Let M be the sub-
YES
set ofN consisting of all those n e N for which there exists a unique Xn e S such that
(n,xn) e R. We shall prove M = N by induction. If 0 t M, then there exists (O,h) e R
with b ¢ a and the set R - {(O,b) } C N X S is in g. Consequently R = n Y
YES
c R - {(O,bJ}, which is a contradiction. Therefore, 0 € M. Suppose inductively that
n E M (that is, (n,xn) € R for a unique xn E S). Then (n + l,fn(xn)) E R also. If
(n + l,c) € R with c ¥- fn(x") then R - {(n + 1 ,c)} e g (verify!), which leads to a
contradiction as above. Therefore, Xn+l = [n(x11) is the unique element of S such
that (n + l,xn+t) e R. Therefore by induction (Theorem 6.1) N = M, whence the

6. THE I NTEGERS 11
assignment n � Xn defines a function 'P : N -4 S with graph R. Since (O,a) e R
we must have ffJ(O) = a. For each n e N, (n,xn) = (n,cp(n)) e R and hence
(n + l,f.n(ffJ(n)) e R since R e g. But (n + 1,xn+l) e R and the uniqueness of Xn+l
imply that '{J(n + 1) = Xn+l = f.n(ffJ(n)). •
If A is a nonempty set, then a sequence in A is a function N -+ A. A sequence is
usually denoted {a0,a1 , • • • } or {a,.};eN or {a;}, where a; e A is the image of i e N.
Similarly a function N* -+ A is also called a sequence and denoted {a1 ,a2 , • • •} or
{a;};eN• or {a;}; this will cause no confusion in context.
Theorem 6.3. (Division Algorithm) Ifa,b, e Z and a r!= 0, then there exists unique
integers q and r such that b = aq + r, and O < r < lal.
SKETCH OF PROOF. Show that the set S = { h - ax I x e Z, b - ax > 0} is
a nonempty subset of N and therefore contains a least element r = b - aq (for some
q e Z). Thus h = aq + r. Use the fact that r is the least element in S to show
0 < r < lal and the uniqueness of q,r. •
We say that an integer a rt= 0 divides an integer b (written a I b) ifthere is an integer
k such that ak = h. If a does not divide h we write a (h.
Definition 6.4. Thepositive integer c ij' said to be the greatest common divisor ofthe
integers al,a2, . . . ' an if·
(I) c I ai for I < i < n;
(2) d e Z and d I ai for I < i < n � d I c.
Theorem 6.5. If a1,a2., • • • , an are integers, not all 0, then (a1,a2, . . . , an) exists.
Furthermore there are integers kt,k2, • • • , kn such that
SKETCH OF PROOF. Use the Division Algorithm to show that the least posi
tive element ofthe nonempty set S = {x1a1 + x�2 + · · · + Xnan I Xi e Z, � Xiai > OJ
•
is the greatest common divisor of ah . . . , an. For details see Shockley [51 ,p.10]. •
The integers a1,az, . . . , an are said to be relatively prime if (a1 ,a2, • • • , an) = I. A
positive integerp > 1 is said to be prime if its only divisors are ± 1 and ±p. Thus ifp
is prime and a e Z, either (a,p) = p (ifp I a) or (a,p) = 1 (ifp .(a).
Theorem 6.6. If a and b are relatively prime integers and a I be, then a I c. Ifp is
prime and p I a1a2· · ·an, then p I ai f
or some i.

12 PREREQU ISITES AND PRELI M I NARIES
SKETCH OF PROOF. By Theorem 6.5 1 = ra + sb, whence c = rae + sbc.
Therefore a I c. The second statement now follows by induction on n. •
Theorem 6.7. (Fundamental Theorem ofArithmetic) Anypositive integer n > 1 may
be written uniquely in the f
orm n = Ptt1P2t2• • •
Pktk, where P• < P2 < · · · < Pk are
primes and ti > 0 f
or all i.
The proof, which proceeds by induction, may be found in Shockley [51, p.l7].
Let m > 0 be a fixed integer. If a,b e Z and m I (a - h) then a is said to be con
gruent to h modulo m. This is denoted by a = h (mod m).
Theorem 6.8. Let m > 0 be an integer and a,b,c,d e Z.
(i) Congruence modulo m is an equivalence relation on the set ofintegersZ, which
.
has precisely m equivalence classes.
(ii) I
f a = b (mod m) and c = d (mod m), then a + c = b + d (mod m) and
ac = bd (mod m).
(iii) Ifab = ac (mod m) and a and m are relatively prime, then b = c (mod m).
PROOF. (i) The fact tl'at congruence modulo m is an equivalence relation is an
easy consequence of the appropriate definitions. Denote the equivalence class of an
integer a by a and recall property (20), which can be stated in this context as:
li = b � a = b (mod m). (20')
Given any a e Z, there are integers q and r, with 0 < r < m, such that a = mq + r.
Hence a - r = mq and a = r (mod m); therefore, a = r by (20'). Since a was ar
bitrary and 0 � r < m, it follows that every equivalence class must be one of
O,i,2,3, . . . , (m -- 1). However, these m equivalence classes are distinct: for if
0 < i < j < m, then 0 < U - i) < m and m t(J - i). Thus i f= j (mod m) and hence
I � j by (20'). Therefore, there are exactly m equivalence classes.
(ii) We are given m I a - b and m I c - d. Hence m divides (a - b) + (c - d)
= (a + c) - (h + d) and therefore a + c = b + d (mod m). Likewise, m divides
(a - b)c + (c - d)h and therefore divides ac - be + cb - db = ac - hd; thus
ac = bd (mod m).
(iii) Since ab = ac (mod m), m I a(b - c). Since (m,a) = 1, m I b - c by Theo
rem 6.6, and thus b = c (mod m). •
7. TH E AXIOM OF CHOICE, ORDER, AND ZORN'S LEMMA
Note. In this section we deal only with sets. No proper classes are involved.
If I � 0 and { Ai I i e /} is a family ofsets such that Ai � 0 for all i e I, then we
would like to know that II Ai � 0. It has been proved that this apparently in-
id
nocuous conclusion cannot be deduced from the usual axioms of set theory (al-
though it is not inconsistent with them - see P. J. Cohen [59]). Consequently we
shall assume
I

7. T H E AXIOM OF CHOICE, ORDER, AND ZORN 'S LEMMA 13
The Axiom of Choice. The product ofa f
amily ofnonempty sets indexed by a non
empty set is nonempty.
See Exercise 4 for another version of the Axiom ofChoice. There are two propo
sitions equivalent to the Axiom ofChoice that are essential in the proofs ofa number
of important theorems. In order to state these equivalent propositions we must
introduce some additional concepts.
A partially ordered set is a nonempty set A together with a relation R on A X A
(called a partial ordering of A) which is reflexive and transitive (see (15), (17) in
section 4) and
antisymmetric : (a,b) e R and (h,a) e R � a = b. (31)
If R is a partial ordering of A, then we usually write a < h in place of (ap) e R. In
this notation the conditions (15), (17), and (31) become (for all a,b,c e A):
a < a·
- '
a < b and b < c =:}
a < b and b < a �
We write a < b if a < b and a ¢ b.
a < c·
- '
a = b.
Elements a,b e A are said to be comparable, provided a < b or b < a. However,
two given elements of a partially ordered set need not be comparable. A partial
ordering of a set A such that any two elements are comparable is called a linear
(or total or simple) ordering.
EXAMPLE. Let A be the power set (set of all subsets) of { l,2,3,4,5 } . Define
C < D if and only if C C D. Then A is partially ordered, but not linearly ordered
(for example, { 1 ,2 } and { 3,4 } are not comparable).
Let (A,<) be a partially ordered set. An element a e A is maximal in A if for every
c e A which is comparable to a, c < a; in other words, for all c e A, a < c � a = c.
Note that if a is maximal, it need not be the case that c < a for all c e A (there may
exist c e A that are not comparable to a). Furthermore, a given set may have many
maximal elements (Exercise 5) or none at all (for example, Z with its usual ordering).
An upper bound of a nonempty subset B of A is an element d e A such that b < d for
every b e B. A nonempty subset B of A that is linearly ordered by < is called a chain
in A.
Zorn's Lemma. IfA is a nonempty partially orderedset such that every chain in A
has an upper hound in A� then A contains a maximal element.
Assuming that all the other usual axioms ofset theory hold, it can be proved that
Zorn's Lemma is true if and only if the Axiom of Choice holds; that is, the two are
equivalent - see E. Hewitt and K. Stromberg [57; p. 14]. Zorn's Lemma is a power
ful tool and will be used frequently in the sequel.
Let B be a nonempty subset ofa partially ordered set (A,<). An element c e B is a
least (or minimum) element of B provided c < b for every b e B. If every nonempty
subset of A has a least element, then A is said to be well ordered. Every well-ordered
set is linearly ordered (but not vice versa) since for all a,h e A the subset { a,b} must

14 PREREQU ISITES AN D PRELI M I NAR I ES
have a least element; that is, a < b or b < a. Here is another statement that can be
proved to be equivalent to the Axiom of Choice (see E. Hewitt and K. Stromberg
[57; p.l4]).
The Well Ordering Principle. IfA is a nonempty set, then there exists a linear
ordering < ofA such that (A,<) is well ordered.
EXAMPLFS. We have already assumed (Section 6) that the set N of natural
numbers is well ordered. The set Z of all integers with the usual ordering by magni
tude is linearly ordered but not well ordered (for example, the subset of negative
integers has no least element). However, each ofthe following is a well ordering of Z
(where by definition a < b � a is to the left of b):
(i) 0,1 ,-1,2,-2,3,- 3, . . . , n,-n, . . . ;
(ii) 0,1 ,3,5,7, . . . , 2,4,6,8, . . . ' - 1 ,-2,-3,-4, . . : ;
(iii) 0,3,4,5,6, . . . t - 1 ,-2,-3,-4, . . . ' 1 ,2.
These orderings are quite different from one another. Every nonzero element a in
ordering (i) has an immediate predecessor (that is an element c such that a is the least
element in the subset { x I c < x} ). But the elements - 1 and 2 in ordering (ii) and - 1
and 1 in ordering (iii) have no immediate predecessors. There are no maximal ele
ments in orderings (i) and (ii), but 2 is a maximal element in ordering (iii). The
element 0 is the least element in all three orderings.
The chief advantage of the well-ordering principle is that it enables us to extend
the principle of mathematical induction for positive integers (Theorem 6.1) to any
well ordered set.
Theorem 7.1. (Principle of Transfinite Induction) If B is a subset ofa well-ordered
set (A,<) such that f
or every a s A,
then B = A.
{ c s A I c < a J C B =} a s B,
PROOF. If A - B rf 0, then there is a least element a s A - B. By the defini
tions of least element and A - B we must have { c s A I c < aJ C B. By hypothesis
then, a s B so that a s B n (A - B) = 0, which is a contradiction. Therefore,
A - B = 0 and A = B. •
EX E R C I SES
1. Let (A,<) be a partially ordered set and B a nonempty subset. A lower bound ofB
is an element d s A such that d < b for every b e B. A greatest lower bound (g.l.b.)
ofB is a lower bound do ofB such that d < do for every other lower bound dofB.
A least upper bound (l.u.b.) of B is an upper bound to of B such that to < t for
every other upper bound t ofB. (A,<) is a lattice if for all a,b e A the set { a,h}
has both a greatest lower bound and a least upper bound.

8. CARDI NAL N U M BERS 15
(a) If S =F- 0, then the power set P(S) ordered by SPt-theoretic inclusion is a
lattice, which has a unique maximal element.
(b) Give an example of a partially ordered set which is not a lattice.
(c) Give an example of a lattice w.ith no maximal element and an example of
a partially ordered set with two maximal elements.
2. A lattice (A, <) (see Exercise l) is said to be complete ifevery nonempty subset of
A has both a least upper bound and a greatest lower bound. A map of partially
ordered sets f: A � B is said to preserve order ifa < a' in A implies /(a) < f(a')
in B. Prove that an order-preserving map fof a complete lattice A into itself has
at least one fixed element (that is, an a s A such that [(a) = a).
3 . Exhibit a well ordering of the set Q of rational numbers.
4. Let S be a set. A choice function for S is a functionffrom the set ofall nonempty
subsets ofS to S such that f(A) s A for all A =F- 0, A C S. Show that the Axiom
of Choice is equivalent to the statement that every set S has a choice function.
5. Let S be the set of all points (x,y) in the plane with y < 0. Define an ordering
by (xi,YI) < (x2,y.,) � x1 = x2 and Y1 < Y2- Show that this is a partial ordering
ofS, and that S has infinitely many maximal elements.
6. Prove that if all the sets in the family { Ai I i e I =F- 0 J are nonempty, then each
of the projections 7rk : II Ai __, Ak is surjective.
i£1
1. Let (A, <) be a linearly ordered set. The immediate successor ofa s A (ifit exists)
is the least element in the set { x s A I a < x } . Prove that if A is well ordered by
< , then at most one element of A has no immediate successor. Give an example
of a linearly ordered set in which precisely two elements have no immediate
successor.
8. CARDINAL NUMBERS
The definition and elementary properties ofcardinal numbers wilJ be needed fre
quently in the sequel. The remainder of this section (beginning with Theorem 8.5),
however, will be used only occasionally (Theorems 11.1.2 and IV.2.6; Lemma V.3.5;
Theorems V.3.6 and VI.1 .9). It may be omitted for the present, if desired.
Two sets, A and B, are said to be equipollent, ifthere exists a bijective map A � B;
in this case we write A � B.
Theorem 8.1. Equipollence is an equivalence relation on the class S of all sets.
PROOF. Exercise; note that 0 � 0 since 0 C 0 X 0 is a relation that is
(vacuously) a bijective function.3 •
Let Io = 0 and for each n E N* let In = { 1 ,2,3, . . . , n } . It is not difficult to prove
that lm and In are equipollent if and only if m = n (Exercise 1). To say that a set A
3See page 6.

16 PREREQU I SITES AN D PRELI M I NARIES
has precisely n elements means that A and In are equipollent, that is, that A and In are
in the same equivalence class under the relation of equipollence. Such a set A (with
A .-...J In for some unique n > 0) is said to be finite ; a set that is not finite is infinite.
Thus, for a finite set A, the equivalence class of A under equipollence provides an
answer to the question: how many elements are contained in A? These considerations
motivate
0
Definition 8.2. The cardinal number (or cardinality) ofa set A, denoted IAJ, is the
equivalence class ofA under the equivalence relation ofequipollence. lAI is an infiniteor
finite cardinal according as A is an infinite orfinite set.
Cardinal numbers will also be denoted by lower case Greek letters: a,f3,'Y, etc.
For the reasons indicated in the preceding paragraph we shall identify the integer
n > 0 with the cardinal number !Inl and write IInl = n, S'? that the cardinal number
of a finite set is precisely the number of elements in the set.
Cardinal numbers are frequently definedsomewhat differently than we have done
so that a cardinalnumber is in fact aset(instead ofa proper class as in Definition 8.2).
We have chosen this definition both to save time and because it better reflects the
intuitive notion of "the number of elements in a set." No matter what definition of
cardinality is used, cardinal numbers possess the following properties (the first two
of which are, in our case, immediate consequences of Theorem 8.1 and Defini
tion 8.2).
(i) Every set has a unique cardinal number;
(ii) two sets have the same cardinal number ifand only if they areequipollent
(IAI = IBI <=> A � B);
(iii) the cardinal number ofafinite set is the number ofelements in the set.
Therefore statements about cardinal numbers are simply statements about equipol
lence of sets.
EXAMPLE. The cardinal number ofthe set N ofnatural numbers is customarily
denoted � (read "aleph-naught"). A set A of cardinality t-to (that is, one which is
equipollent to N) is said to be denumerable. The set N*, the set Z ofintegers, and the
set Q ofrational numbers are denumerable (Exercise 3), but tpe set R ofreal numbers
is not denumerable (Exercise 9).
Definition 8.3. Let a and{j be cardinalnumbers. The sum a + {3 is defined to be the
cardinal number lA U Bl, where A and B are disjoint sets such that IAI = a and
IBI = {j. The product a{3 is defined to be the cardinal number lA X Bl.
It is not act�ally necessary for A and B to be disjoint in the definition of the
product a{j (Exercise 4). By the definition of a cardinal number a there always
exists a set A such that lAI = a. It is easy to verify that disjoint sets, as required for
the definition ofa + {j, always exist and that the sum a + {3 and product a{3 are in
dependent of the choice of the sets A,B (Exercise 4). Addition and multiplication of
cardinals are associative and commutative, and the distributive laws hold (Exercise
5). Furthermore, addition and multiplication of finite cardinals agree with addition

__
__
__
__
__
__
__
__
__
__
_
B
_
. _C
_
A
__
R_
D_
I N
_
A
_
L
__
N
_
U
_
M
_
B
_
E
_
R
_
S
__
__
__
__
__
__
__
__
__
___
17 �
and multiplication of the nonnegative integers with which they are identified; for if
A has m elements, B has n elements and A n B = 0, then A U B has m + n ele
ments and A X B has mn elements (for more precision, see Exercise 6).
Definition 8.4. Let a,{3 be cardinalnumbers andA,B setssuch that IAI = a, IBI = {3.
a is-llss than or equal to {3, denoteda < {3 or{3 > a, ifA is equipollent with a subset of
B (that is, there is an injective map A � B). a is strictly less than {3, denoted a < {3
or {3 > a, ifa < {3 and a "# {3.
It is easy to verify that the definition of < does not depend on the choice of A
and B (Exercise 7). It is shown in Theorem 8.7 that the class of all cardinal numbers
is linearly ordered by < . For finite cardinals < agrees with the usual ordering ofthe
nonnegative integers (Exercise 1). The fact that there is no largest cardinal number is
an immediate consequence of
Theorem 8.5. IfA is a set and P(A) its power set, then IAI < IP(A)I.
SKETCH OF PROOF. The assignment a � (a} defines an injective map
A � P(A) so that lA I < IP(A)j. If there were a bijective map f: A � P(A), then for
some ao e A, f(ao) = B, where B = {a e A I a t f(a)} C A. But this yields a con
tradiction: ao e B and ao + B. Therefore fAI � IP(A)I and hence lAI < jP(A)I. •
REMARK. By Theorem 8.5, No = INI < IP(N)I. It can be shown that
jP(N)I = IRI, where R is the set of real numbers. The conjecture that there is no
cardinal number {3 such that �o < {3 < jP(N)I = IRI is called the Continuum Hy
pothesis. It has been proved to be independent of the Axiom of Choice and of the
other basic axioms of set theory; see P. J. Cohen [59].
The remainder of this section is devoted to developing certain facts that will be
needed at several points in the sequel (see the first paragraph of this section).
Theorem 8.6. (Schroeder-Bernstein) If A and B are sets such that IAI < IBI and
IBf < IAI, then IAI = IBI.
SKETCH OF PROOF. By hypothesis there are injective maps f: A � B and
g : B � A. We sl:all use fand g to construct a bijection h : A --+ B. This will imply
that A � B and hence !AI = IBI. If a e A, then since g is injective the set g-1(a) is
either empty (in which case we say that a is parenf[ess) or consists ofexactly one ele
ment b e B (in which case we write g-1(a) = b and say that b is the parent of a).
Similarly for b e B, we have either f-1(b) = 0 (b is parent/ess) or f-1(b) = a' E A
(a' is theparent of b). If we continue to trace back the "ancestry" ofan element a E A
in this manner, one of three things must happen. Either we reach a parentless ele
ment in A (an ancestor of a e A), or we reach a parentless element in B (an ancestor

18 PREREQUISITES AND PRELIM INARI ES
of a), or the ancestry of a e A can be traced back forever (infinite ancestry). Now de
fine three subsets of A [resp. B] as follows:
At = {a e A I a has a parentless ancestor in A } ;
A2 = { a e A I a has a parentless ancestor in B } ;
Aa = { a e A I a has infinite ancestry} ;
Bt = { b e B I b has a parentless ancestor in A } ;
B2 = { b e B I b has a parentless ancestor in B} ;
Ba = {b e B I b has infinite ancestry} .
Verify that the Ai [resp. Bi] are pairwise disjoint, that their union is A [resp. B]; that
fl Ai is a bijection A. � Bi for i = 1 , 3; and that g I B2 is a bijection B2 � A2. Con
sequently the map h : A � B given as follows is a well-defined bijection:
Theorem 8.7. The class of all cardinal numbers is linearly orderedby < . Ifa and {j
are cardinal numbers, then exactly one ofthe f
ollowing is true:
a < {j; a = {3; {3 < a (TrichotomyLaw).
SKETCH OF PROOF. It is easy to verify that < is a partial ordering. Let a,{3
be cardinals and A,B be sets such that lAI = a, IBI = {j. We shall show that < is a
linear ordering (that is, either a < {3 or {3 < a) by applying Zorn's Lemma to the set
5' of all pairs (f,X), where X c A and f: X -t B is an injective map. Verify that
5' � 0 and that the ordering of 5" given by (ft,Xt) < (f2,XJ if and only ifX1 C X2
and 12 1 X1 = h is a partial ordering of 5'. If { {fi,Xi) I i e /} is a chain in 5", let
X = U Xi and define f: X -t B by f(x) = fi(x) for x €Xi. Show that fis a well-de-
w
.
fined injective map, and that (f,X) is an upper bound in 5' of the given chain. There
fore by Zorn's Lemma there is a maximal element (g,X) of 5". We claim that either
X = A or Im g = B. For ifboth ofthese statements were false we could find a E A - X
and b e B - Im g and define an injective map h : X U fa} � 8 by h(x) = g(x) for
x e X and h(a) = b. Then (h,X U { a) ) E g: and (g,X) < (h,X U {a I), which contra
dicts the maximality of(g,X). Therefore either X = A so that lA I < IBI or Im g = B
-I
in which case theinjective map B �X c A shows that IBI < !AI. Use these facts, the
Schroeder-Bernstein Theorem 8.6 and Definition 8.4 to prove the Trichotomy
Law. •
REMARKS. A family of functions partially ordered as in the proof of Theorem
8.7 is said to be ordered by extension. The proof of the theorem is a typical example
of the use of Zorn's Lemma. The details of similar arguments in the sequel will fre
quently be abbreviated.
Theorem 8.8. Every infinite set has a denumerable subset. In particular, Nu < a for
every infinite cardinal number a.
I
'
I

8. CARDI NAL N U M BERS 19
SKETCH OF PROOF. IfB is a finite subset ofthe infinite set A, then A - B is
nonempty. For each finite subset B of A, choose an element XB E A - B (Axiom of
Choice). Let F be the set of all finite subsets of A and define a map f: F � F by
f(B) = B U { xn} . Choose a E A. By the Recursion Theorem 6.2 (with In = f for
all n) there exists a function cp : N � F such that
<P(O) = {a } and <{J(n + 1) = f(<P(n)) = <P(n) U { x�<nd (n > 0).
Let g : N � A be the function defined by
g(O) = a; g(l) = Xcp(O) = X(a} ; • . • ; g(n + 1) = X�<n> ; • • • •
Use the order properties of N and the following facts to verify that g is injective:
(i) g(n) E <P(n) for all n > 0;
(ii) g(n) f cp(n - 1) for all n > 1 ;
(iii) g(n) t cp(m) for all m < n.
Therefore Im g is a subset of A such that lim gl = INI = �o. •
Lemma 8.9. IfA is an infinite set and F afinite set then lA U Fl = IAI. Inparticular,
a + n = a f
or every infinire cardinal nutnber a and ecery natural nutnber (finite
cardinal) n.
SKETCH OF PROOF. It suffices to assume A n F = 0 (replace F by F - A
if necessary). IfF = { b1,b2, . . . , bn } and D = t Xi I i E N* f is a denumerable subset of
A (Theorem 8.8), verify that f: A � A U F is a bijection, where /is given by
bi for x = xi, 1 < i < n;
f(x) = Xi-n for x = Xi, i > n;
X for x E A - D. •
Theorem 8.10. Ifa and {3 are cardinal nu1ubers such that {3 < a and a is infinite,
then a + {3 = a.
SKETCH OF PROOF. It suffices to prove a + a = a (simply verify that
a < a + {3 < a + a = a and apply the Schroeder-Bernstein Theorem to conclude
a + {3 = a). Let A be a set with !AI = a and let g: be the set of all pairs (f,X),
where X C A and f: X X { 0,1 } � X is a bijection. Partially order 5 by extension
(as in the proofofTheorem 8.7) and verify that the hypotheses ofZorn"s Lemma are
satisfied. The only difficulty is showing that 5 -;e 0. To do this note that the map
N X { 0,1 } � N given by (n,O) � 2n and (n,l) � 2n + 1 is a bijection. Use this fact
to construct a bijection f: D X { 0,1 } ----) D, where D is a denumerable subset of A
(that is, IDI = INI ; see Theorem 8.8).Therefore by Zorn's Lemma there is a maximal
element (g,C) E 5.
Clearly Co = { (c,O) I c E CJ and C1 = { {c,l) I c E C} are disjoint sets such that
!Col = ICI = I Cll and c X f O,l } = Co u cl. The map g : c X { 0,1 J � c is a bi
jection. Therefore by Definition 8.3 ,
I CI = !C X l 0,1 J I = ICo U Ctl = !Col + IC1I = IC! + ICI.

20 PREREQUISITES AN D PRELI M I NARIES
To complete the proof we shall show that I Cl = a. If A - C were infinite, it would
contain a denumerable subset B by Theorem 8.8, and as above, there would be a bi
jection t : B X { 0,1 } � B. By combining t with g, we could then construct a bijec
tion h : (C U B) X t 0,1 } � C U B so that (g,C) < (h,C U B) E g:, which would
contradict the maximality of (g,C). Therefore A - C must be finite. Since A is in
finite and A = C U (A - C), C must also be infinite. Thus by Lemma 8.9, ICI =
JC U (A - C)l = JAI = a. •
Theorem 8.1 1. I
f a and p are cardinal numbers such that 0 =I= p < a and a is
infinite. then aP = a; in particular. aN0 = a and ifP isfinite Noll = N0•
SKETCH OF PROOF. Since a < a{j < aa it suffices (as in the proofofTheo
rem 8.10) to prove aa = a. Let A be an infinite set with IAI = a and let g: be the set
of all bijections f: X X X � X, where X is an infinite subset of A. To show that
g: � 0, use the facts that A has a denumerable subset D (so that IDI = IN! = IN*I)
and that the map N* X N* � N* given by (m,n) � 2m-
1(2n - 1 ) is a bijection.
Partially order g: by extension and use Zorn's Lemma to obtain a maximal element
g : B X B � B. By the definition ofg, IBI IBI = IB X Bl = IBI . To complete the proof
we shall show that IBI = lAI = a.
Suppose lA - BJ > IBJ . Then by Definition 8.4 there is a subset C ofA - B such
that I CI = IBI. Verify that ICI = IBI = I B X Bl = IB X Cl = I C X Bl = IC X Cj
and that these sets are mutually disjoint. Consequently by Definition 8.3 and Theo
rem 8.10 j(B U C) X (B U C)j = J(B X B) U (B X C) U (C X B) U (C X C)J
= IB X Bl + IB X Cj + J C X Bl + J C X Cl = (JBI + IBI) + (JCI + ICI) = IBI +
ICJ = IB U Cl and there is a bijection (B U C) X (B U C) � (B U C), which con
tradicts the maximality ofg in g:. Therefore� by Theorems 8.7 and 8. 10 IA -
B I � I B I
and IBI = lA - Bl + IBI = j(A - B) U Bl = !AI = a. •
Theorem 8.12. Let A be a set andfor each integer n > 1 /er An = A X A X · · · X A
(n f
actors).
(i) IfA isfinite, then IAnj = lAin, and ifA is infinite, then IAni = IAJ.
(ii) J U Anj = NoJAJ.
nEN*
SKETCH OF PROOF. (i) is trivial if JA] is finite and may be proved by induc
tion on n if IAI is infinite (the case n = 2 is given by Theorem 8.1 1 ). (ii) The sets
An (n > I) are mutually disjoint. If A is infinite, then by (i) there is for each n a bijec
tion fn : An � A. The map U An � N* X A, which sends u € An onto (n,fn(u)), is a
neN*
bijection. Therefore I U Anj = IN* X AI = IN*I IAI = t-toiA!. (ii) is obviously true
nEN*
if A = 0 . Suppose, therefore, that A is nonempty and finite. Then each An is non-
empty and it is easy to show that t-to = IN*I < I U Anj. Furthermore each A11 is
nEN*
finite and there is for each n an injective map gn : An � N*. The map U An �
nEN*
N* X N*, which sends u e An onto (n,g,Ju)) is injective so that I U Ani < IN* X N*l
nEN*
= IN*I = No by Theorem 8.1 1 . Therefore by the Schroeder-Bernstein Theorem
I U Ani = t-ta. But No = NoiAI since A is finite (Theorem 8.1 1 ). •
nEN*

8. CARDI NAl NUM BERS 21
Corollary 8.13. IfA is an infinite set and F(A) the set ofallfinite subsets ofA, then
jF(A)I = IAI.
PROOF. The map A � F(A) given by a � { a} is injective so that iAI < IF(A)I.
For each n-element subset s ofA' choose (a., . . . ' an) e An such thats = t Gt , . . . ' an } .
This defines an injective map F(A) � U An so that IF(A)I < I U An_l = �oiAI = !AI
nEN* nEN*
by Theorems 8.1 1 and 8.12. Therefore, !AI = IF(A)I by the Schroeder-Bernstein
Theorem 8.6. •
EX E R C I SES
1 . Let Io = 0 and for each n e N* let In = t 1 ,2,3, . . . , n} .
(a) In is not equipollent to any ofits proper subsets [Hint: induction].
(b) Im and I71 are equipollent if and only if m = n.
(c) lm is equipollent to a subset of In but In is not equipollent to any subset of Im
if and only if m < n.
2. (a) Every infinite set is equipollent to one of its proper subsets.
(b) A set is finite if and only if it is not equipollent to one of its proper subsets
[see Exercise 1].
3. (a) Z is a denumerable set.
(b) The set Q of rational numbers is denumerable. [Hint: show that
IZI < IQI < IZ X Zl = IZI.]
4. IfA,A',B,B' are sets such that !AI = lA'I and IBI = IB'I, then lA X Bl = lA' X B'l.
If in addition A n B = 0 = A' n B', then lA U Bl = lA' U B'j. Therefore
multiplication and addition of cardinals is well defined.
5. For all cardinal numbers a,{3;y :
(a) a + {3 = {3 + a and a{3 = {3a (commutative laws).
(b) (a + {3) + 'Y = a + ({3 + ")') and {af3)"Y = a(f3'Y) (associative laws).
(c) a({3 + ")') = a{3 + a')' and {a + f3)'Y = a')' + f3'Y (distributive laws).
(d) a + 0 = a and al = a.
(e) If a ¢ 0, then there is no {3 such that a + {3 = 0 and if a ¢ 1, then there is
no {3 such that a{3 = l . Therefore subtraction and division of cardinal num
bers cannot be defined.
6. Let In be as in Exercise 1. If A � Im and B "'-./ In and A n B = 0, then (A U B)
� Im+n and A X B r-....; Imn· Thus if we identify lAI with m and IBI with n, then
IAI + IBI = m + n and IAIIBI = mn.
7. If A � A', B "" B' and f: A � B is injective, then there is an injective map
A' ---4 B'. Therefore the relation < on cardinal numbers is well defined.
8. An infinite subset of a denumerable set is denumerable.
9. The infinite set of real numbers R is not denumerable (that is, No < IRI). [Hint:
it suffices to show that the open interval (0,1) is not denumerable by Exercise 8.
You may assume each real number can be written as aninfinite decimal. If (0,1) is
denumerable there is a bijectionf: N* � (0,1). Construct an infinite decimal (real
number) .a.az· · · in (0�1) such that an is not thenth digit in the decimalexpansion
of f(n). This number cannot be in Im /.]

22 PREREQU ISITES AN D PRELI M I NARI ES
10. Ifa,{j are cardinals, define aP to be the cardinal number ofthe set ofall functions
B ---+ A, where A,B are sets such that IAI = a, IBI = {j.
(a) aB is independent of the choice of A,B.
(b) aB+-r = (aB)(a'Y); (a{j)'Y = (a'Y)(B'Y); aP-r = (afJ}'Y.
(c) If a � {3, then a'Y < {3'Y.
(d) Ifa,[J are finite with a > I , fJ > 1 and y is infinite, then a_Y = {JY.
(e) For every finite cardinal n, an = aa · · · a (n factors). Hence an = a if a is
infinite.
(0 If P(A) is the power set of a set A, then IP(A)I = 21AI.
1 1. If I is an infinite set, and for each i e I Ai is a finite set, then IU Ail < JiJ.
i£1
12. Let a be a fixed cardinal number and suppose that for every i e I, Ai is a set with
IAsl ;:: a. Then I U A,l < lila.
itl
r
.

CHAPTER I
GROU PS
The concept ofa group is offundamental importance in the study ofalgebra. Groups
which are, from the point ofview of algebraic structure, essentially the same are said
to be isomorphic. Ideally the goal in studying groups is to classify all groups up to
isomorphism, which in practice means finding necessary and sufficient conditions for
two groups to be isomorphic. At present there is little hope of classifying arbitrary
groups. But it is possible to obtain complete structure theorems for various restricted
classes of groups, such as cyclic groups (Section 3), finitely generated abelian
groups (Section 11.2), groups satisfying chain conditions (Section 11.3) and finite
groups of small order (Section 11.6). In order to prove even these limited structure
theorems, it is necessary to develop a large amount of miscellaneous information
about the structure of (more or less) arbitrary groups (Sections 1 , 2, 4, 5, and 8 of
Chapter I and Sections 4 and 5 ofChapter II). In addition we shall study some classes
of groups whose structure is known in large part and which have useful applications
in other areas of mathematics, such as symmetric groups (Section 6), free [abelian]
groups (Sections 9 and 11.1), nilpotent and solvable groups (Sections II.7 and 11.8).
There is a basic truth that applies not only to groups but also to many other
algebraic objects (for example, rings, modules, vector spaces, fields): in order to
study effectively an object with a given algebraic structure, it is necessary to study as
well the functions that preserve the given algebraic structure (such functions are
called homomorphisms). Indeed a number of concepts that are common to the
theory of groups, rings, modules, etc. may b� described completely in terms of ob
jects and homomorphisms. In order to provide a convenient language and a useful
conceptual framework in which to view these common concepts, the notion of a
category is introduced in Section 7 and used frequently thereafter. Of course it is
quite possible to study groups, rings, etc. without ever mentioning categories. How
ever, the small amount ofeffort needed to comprehend this notion now will pay large
dividends later in terms of increased understanding of the fundamental relationships
among the various algebraic structures to be encountered.
With occasional exceptions such as Section 7, each section in this chapter de
pends on the sections preceding it.
23

24 CHAPTER I GROU PS
1. SEMIGROUPS, MONOI DS AN D GROU PS
If G is a nonempty set, a binary operation on G is a function G X G � G. There
are several commonly used notations for the image of(a,b) under a binary operation:
ab (multiplicative notation), a + b (additive notation), a· b, a * b, etc. For con
venience we shall generally use the multiplicative notation throughout this chapter
and refer to ab as the product of a and b. A set may have several binary operations
defined on it (for example, ordinary addition and multiplication on Z given by
(a,b) � a + b and (a,b) J----; ab respectively).
l
Definition 1.1. A semigroup is a nonempty set G together with a binary operation
on G which is I
(i) associative: a(bc) = (ab)c f
or all a, b, c E G;
a monoid is a semigroup G which contains a
(ii) (two-sided) identity element e c: G such that ae = ea = a for all a E G.
A group is a monoid G such that
(iii) for every a E G there exists a (two-sided) inverse element a-l E G such that r
a-•a = aa-1 = e. '
A semigroup G is said to be abelian or commutative ifits binary operation is
(iv) commutative: ab = ba for all a,b E G.
Our principal interest is in groups. However, semigroups and monoids are con
venient for stating certain theorems in the greatest generality. Examples are given
below. The order of a group G is the cardinal number I GI . G is said to be finite
lresp. infinite] if l Gl is finite [resp. infinite].
Theorem 1.2. JfG is a monoid, then the identity element e is unique. IfG is a group,
then
(i) c c: G and cc = c => c = e;
(ii) f
or all a, b, c E G ab = ac =::) b = c and ba = ca => b = c (left and right·
cancellation);
(iii) f
or each a E G, the inverse element a-• is unique;
(iv) f
or each a E G, (a-•)-1 = a;
(v) f
or a, b E G, (ab)-
1 = b-Ia-•;
(vi) for a, b E G the equations ax = b and ya = b have unique solutions in
G : x = a-•b and y = ba-•.
SKETCH OF PROOF. If e' is also a two-sided identity, then e = ee' = e'.
(i) cc = c ::=:} c-1(cc) = c-•c � (c-1c)c = c-1c =::) ec = e =::) c = e; (ii), (iii) and (vi)
are proved similarly. (v) (ab)(b-1a-1) = a(bb-1)a-1 = (ae)a-1 = aa-1 = e =::) (ab)-1
= b-1a-1 by (iii) ; (iv) is proved similarly. •
;
t
I
It
1
I'
l

1. SEM I G ROU PS, MONOIDS A N D GROU PS 25
If G is a n1onoid and the binary operation is written multiplicatively, then the
identity elen1ent of G will always be denoted e. If the binary operation is written
additively, then a + b (a, b s G) is called the sum of a and b, and the identity element
is denoted 0; if G is a group the inverse of a 2 G is denoted by -a. We write a - b
for a + ( - h). Abelian groups are frequently written additively.
The axion1s used in Definition 1 . 1 to define a group can actually be weakened
considentbly.
Proposition 1.3. Let G be a sentigroup. Tlu:n G is a group ifand only ifthefollowing
conditions hold:
(i) there exists an elenu?nt e € G such that ea = a for all a € G (left identity
e/entent);
(ii) f
or each a c G, there exists an ele1nent a-
1 € G such that a-1a = e (left incerse).
RE:1ARK. An analogous result holds for "right inverses" and a "right identity.·�
SKETCH OF PROOF OF 1 .3. (�) Trivial. (<=) Note that Theorem 1 .2(i) is
true under these hypotheses. G � 0 since e € G. If a € G, then by (ii) (aa-•)(aa-1)
= a(a-1a)a-1 � a(ea-1) = aa-1 and hence aa-1 = e by Theorem 1 .2(i). Thus a-1 is a
two-sided inverse of a . Since ae = a(a-1a) = (aa-1)a = ea = a for every a € G, e is a
two-sided identity. Therefore G is a group by Definition 1.1 . •
Proposition 1.4. Let G he a sen1igroup. Then G is a group ifand only iff
or all
a, b € G the equations ax = b and ya = b hace solutions in G.
PROOF. Exercise; use Proposition 1 .3. •
EXAitiPLES. The integers Z, the rational numbers Q, and the real numbers R
are each infinite abelian groups under ordinary addition. Each is a monoid under
ordinary multiplication. but not a group (0 has no inverse). However, the nonzero
elements of Q and R respectively form infinite abelian groups under multiplication.
The even integers under multiplication form a semigroup that is not a monoid.
EXAMPLE. Consider the square with vertices consecutively numbered 1 ,2,3,4 ,
center at the origin of the x-y plane. and sides parallel to the axes.
y
1 2
X
4 3
Let D4* be the following set of utransformations" of the square. D4* =
{ R,R2,R3,/,Tx,Ty,T1•3,T2•4 } , where R is a counterclockwise rotation about the center of
90°, R2 a counterclockwise rotation of 180°, R3 a counterclockwise rotation of 270°

26 CHAPTER I G ROU PS
and I a rotation of 360° ( = 0°) ; Tx is a reflection about the x axis, T1,3 a reflection
about the diagonal through vertices 1 and 3; similarly for Ty and· T2,4· Note that
each U e D4* is a bijection of the square onto itself. Define the binary operation in
D4* to be composition offunctions: for U,V e D4*, U o Vis the transformation Vfol
lowed by the transformation U. D4* is a nonabelian group of order 8 called the group
of symmetries of the square. Notice that each symmetry (element of D4*) is com
pletely determined by its action on the vertices.
EXAMPLE. Let S be a nonempty set and A(S) the set of all bijections S � S.
Under the operation of composition of functions, fo g, A(S) is a group, since com
position is associative, composition of bijections is a bijection, 1 s is a bijection, and
every bijection has an inverse (see (13) of Introduction, Section 3). The elements of
A(S) are caJied permutations and A(S) is called the group of permutations on the
set S. IfS = { 1 ,2,3, . . . , n } , then A(S) is called the symmetric group on n letters and
denoted S.n. Verify that ISnl = n! (Exercise 5). The groups s71 play an important
role in the theory of finite groups.
Since an element u ofSn is a function on the finite set S = { 1,2, . . . , n } , it can be
described by listing the elements ofS on a line and the image ofeach element under u
directly below it: (� � � �)- The product UT of two elements ofs71 is the
h l2 Ia In
composition function rf
ollowedby u; that is, the function onS given by k � u(r(k)).1
For instance, let u = G i ; :)and r = (! i ; :)be elements ofs•. Then
under ur, I � u(r(I)) = u(4) = 4, etc. ; thus ur = G i ; :)(! i ; :)
= (! � � �); similarly, ru = (! i ; �)G i ; :) =
G � � �)
This example also shows that Sn need not be abelian.
Another source of examples is the following method of constructing new groups
from old. Let G and H be groups with identities eG, ell respectively, and define the
direct product of G and H to be the group whose underlying set is G X H and whose
binary operation is given by:
(a,b)(a',b') = (aa',bb'), where a,a' e G; h,b' e H.
Observe that there are three different operations in G, H and G X H involved in this
statement. It is easy to verify that G X H is, in fact, a group that is abelian if both G
and H are; (eG,en) is the identity and (a-1,b-1) the inverse of (a,b). Clearly IG X HI
= IGi lHI (Introduction, Definition 8.3). If G and H are written additively, then we
write G EB H in place of G X H.
Theorem 1.5. Let R ("'-') be an equit�alence relation on a monoid G such that a. ""' a2
and b1 ,__, b2 ilnply a1b1 "'-' a2b2 for all ai,bi e G. Then the set G/R ofall eq"!jva/ence
cla�-ses ofG under R is a monoid under the binary operation defined by (a)(b) = ab,
wllerexdenotes theequivalenceclass ofx E G. IfG is an [abelian] group, then so isGjR.
1In many books, however, the product ur is defined to be "
u followed by r .
'·

1. SEM I G ROU PS, MON O I DS AND GROU PS •
An equivalence relation on a monoid G that satisfies the hypothesis of the theo
renl is called a congruence relation on G.
PROOF OF 1.5. If a. = a2 and h. = b2 (ai, bi e G), then a1 1'-J a2 and bt 1'-J b2 by
(20) of Introduction, Section 4. Then by hypothesis a.b. ""' a2b2 so that a1b1 = a2b2
by (20) again. Therefore the binary operation in GIR is well defined (that is, inde
pendent of_!!le choice of equivalence class representatives). It is associative since
a(h c) = a(bc) = a(bc) = (ab)c = (ab)c = (a b)c. e is the identity element since
(a)(e) = ae = a = ea = (e)(li). Therefore GIR is a monoid. If G is a group, then
li E GIR clearly has inverse a-
1
so that GIR is also a group. Similarly, G abelian im
plies GIR abelian. •
EXAMPLE. Let m be a fixed integer. Congruence modulo m is a congruence re
lation on the additive group Z by Introduction, Theorem 6.8. Let Zm denote the set
of equivalence classes of Z under congruence modulo m. By Theorem 1 .5 (with addi
tive notation)Zm is an abelian group, with addition given by li + b = a + b(a,b e Z).
The proof of Introduction, Theorem 6.8 shows that Zm = {0,1, . . . , m - 1 } so that
Zm is a finite group of order m under addition. Zm is called the (additive) group of
integers modulo m. Similarly since Z is a commutative monoid under multiplication,
and congruence modulo m is also a congruence relation with respect to multiplica
tion (Introduction, Theorem 6.8), Zm is a commutative monoid, with multiplication
given by (a)(b) = {i/j (a,b e Z). Verify that for all a, b, c eZm:
a(b + c) = ab + lie and (a + b)c = lie + be (distributivity).
Furthermore if p is prime, then the nonzero elements of Zp form a multiplicative
group of order p - 1 (Exercise 7). It is customary to denote the elements of Zm as
0,1, . . . , m - 1 rather than 0,1, . . . , m - 1 . In context this ambiguous notation
will cause no difficulty and will be used whenever convenient.
EXAMPLE. The following relation on the additive group Q of rational numbers
is a congruence relation (Exercise 8):
a 1'-J b ¢:==� a - b e Z.
By Theorem 1 .5 the set of equivalence classes (denoted Q/Z) is an (infinite) abelian
group, with addition given by a + b = a + b. Q/Z is called the group of rationals
modulo one.
Given a., . . . , a71 e G (n > 3) it is intuitively plausible that there are many ways
of inserting parentheses in the expression ala2 . . . an so as to yield a "meaningful"
product in G of these n elements in this order. Furthermore it is plausible that any
two such products can be proved equal by repeated use of the associative law. A
necessary prerequisite for further study of groups and rings is a precise statement
and proof of these conjectures and related ones.
Given any sequence ofelements of a semigroup G, {a�,a2 . . . 1 define inductively a
meaningful product of a1, • • • , an (in this order) as follows. If n = 1 , the only mean
ingful product is a1 . If n > 1 , then a meaningful product is defined to be any product
Of the form (at • · · a,.)(am+l · · · an) Where m < n and (aJ · · • am) and (am+I • • · a,,) are
meaningful products of m and n - m elements respectively.2 Note that for each
2To show that this definition is in fact well defined requires a stronger version of the
Recursion Theorem 6.2 of the Introduction; see C. W. Burrill [56; p. 57}.

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